enumeration under group action: unsolved graphical enumeration problems, iv

JOURNAL OF COMBINATORIAL THEORY 8, 1--11 (1970)

Enumeration under Group Action: Unsolved Graphical Enumeration Problems, IV*

FRANK HARARY**

Department of Mathematics, University of Michigan, Ann Arbor, Michigan 48104

Received November 29, 1968

ABSTRACT

This review article presents three methods for solving enumeration problems which can be construed as the determination of the number of orbits of an appropriate permutation group. Such a group must be constructed in accordance with the idio- syncrasies of the configurations to be counted and the equivalence relation on them. Thus three different binary operations on permutation groups, the sum, product, and power group, are defined and the structure of each is investigated. Applications of the corresponding theorems for enumeration under group action are provided. We conclude with a table of 27 current unsolved problems in graphical enumeration.

1. INTRODUCTION

There is a considerable interest in graphica l enumera t ion prob lems , but apparen t ly a major i ty o f otherwise bri l l iant g raph theorists canno t count. W h y ? Perhaps it is because the combina to r i a l aspects o f enumera t ion are a bi t more subtle than the convent ional methods o f p r o o f used in s t ructura l g raph theory. I t m a y be the presence o f pe rmu ta t i on groups and the concomi tan t a lgebra ic formulas and viewpoints. A closely related and somewhat equivalent a p p r o a c h using general ized M6b ius inversion fo rmulas (see R o t a [13]), for enumera t ion o f configurat ions by studying associa ted lattices, does no t seem to make count ing p rob lems any more pa la tab le to the world o f g raph theorists.

In an effort to make some o f the s t andard enumera t ion methods avai lable , we deve lop in this art icle three enumera t ion theorems, a l ready exist ing in the l i terature, in a unified presenta t ion. Each o f these is asso-

* Presented at the Yale University Conference on Combinatorial Theory in honor of Professor Oystein Ore (May, 1968).

** Research supported in part by a grant from the U. S. Air Force Office of Scientific Research. This paper was prepared with the considerable assistance of Paul Stockmeyer and Dennis Geller.

1 �9 1970 by Academic Press, Inc.

2 HARARY

ciated with an operation on permutation groups: the sum, product, and power of two groups. These theorems are illustrated by finding the number of necklaces of certain types.

For solving new enumeration problems, surely additional combinations of permutation groups will need to be devised. One such operation, for example, is that discovered by Palmer and Robinson [10], called the matrix group of two groups. A list of 27 unsolved combinatorial enumeration problems involving graphs and related structures concludes this article.

2. THE THEOREMS OF BURNSIDE AND POLYA

The fundamental result in the theory of enumeration under group action is the theorem of Burnside [2, p. 191], which counts the number of orbits of a permutation group. Most graphical enumeration problems are con- cerned only with the number of non-isomorphic graphs of a given type. Thus a typical solution consists of first finding a permutation group whose orbits are exactly the isomorphism classes of graphs under consideration, and then applying an appropriate generalization of Burnside's Theorem.

Let A be a permutation group acting on a set X. For each ~ ~ A, we definejk(~) as the number of cycles of length k in the disjoint cycle decom- position of ~. In particular, jl(~) is the number of elements of X fixed by a.

BURNSIDE'S THEOREM. group A is given by

The number N(A) of orbits of the permutation

1 N(A) = I A - ~ ~ .~(cx). (1)

~x~A

As a typical application of this theorem, we will count the number of non-equivalent necklaces containing n beads, each of which has one of m colors. Two necklaces are considered equivalent if one can be rotated or reflected onto the other. Casting this problem in more mathematical terms, we will count the number of equivalence classes of functions whose domain is the set of n places on the framework of a necklace and whose range is the set of m colors. Two functions f and g are then equivalent if there exists a permutation ~ in the dihedral group D~, acting on the places in the framework, such tha t f (x ) = g(ax). We can now define a permutation group /~i induced by D~ and acting on these functions f by Of(x) = f (~c ) where ~ ~ D , and (~ is the corresponding element o f / ) ~ . Of course D , also depends on the number m of colors. Then the equiva-

U N S O L V E D G R A P H I C A L ENUMERATION PROBLEMS, IV 3

lence classes of functions are exactly the orbits o f / ~ n . By Burnside's Theorem, the number N(/~n) of necklaces on n places is given b y

1 = Z

c ~ D n

For a specific example, take n = 5, m = 2. Thus we are counting necklaces with 5 beads of two colors, say black and white. The identity element in f i 5 (which of course depends also on the value m = 2) fixes all 2 ~ functions, the four permutations that correspond to rotations of the framework each fix only 2 functions, and the five permutations corresponding to reflections each fix 2 3 functions. Thus the number N(/)s) of necklaces with 5 places and 2 colors is given by

N(/~5) = 1 (25 q_ 4 �9 21 -~- 5 �9 2 3) = 8.

These 8 necklaces are illustrated in Figure 1, where the white beads are the points with open dots.

FIGURE 1

The major weakness of Burnside's Theorem is that it does not easily yield the number of necklaces with a given number of beads of each color; more generally, it does not give the number of "configurations" with a given "weight." A powerful generalization which handles such questions was developed by P61ya [11] by exploiting generating functions.

We first define the cycle index Z(A) of a permutation group A of degree d as the polynomial in d variables al ..... aa given by

1 a ,7 ~'~(~) Z(A)---- f A--S Z �9 (2)

a ~ A k = l

For any permutation ~ ~ A, the numbers.]k = Jk(o0 satisfy

Jx ~- 2j2 + "'" q- dj, = d, (3)

4 HARARY

and constitute a partition of the integer d. I t is useful to employ the vector notation

(J) = 01 .. . . . J~)

to express the cycle structure of a. The general form of a counting problem to which P61ya's Theorem

applies is similar to that for Burnside's Theorem. Not only is there given a domain D and a range R, but in addition a weight function w defined on R. In a typical application, w might assign to each r e R an ordered pair w(r) = (wlr, w2r) of non-negative integers. This leads to a statement of P61ya's Theorem displaying just two variables x and y, a convenient special case. Once again, the objects to be counted are functions from D into R. A group A acting on D is given, and two functions f and g are called A-equivalent if there exists e~ e A such that f (d) = g(ad). Thus A induces a permutation group A on R D whose orbits are the equivalence classes of functions.

W e assign another kind of weight W ( f ) to each f u n c t i o n f e R D by

W(f) = 1-I x~Ol*(a)Y w~*(a)" (4) d e D

Each function in a given orbit of A then has the same weight, so the weight of an orbit can be defined as the weight of any function in it.

Suppose the number of range elements of weight (m, n) is c . . . . and the number of orbits in R ~ with weight xmy ~ is C~.,n �9 Thefigure counting series is

c(x, y) = ~, cm,,x~y"; (5)

it enumerates the elements of R by weight. Similarly the configuration counting series

C(x, y) = E Cm,',xmY" (6)

is the generating function for equivalence classes. Finally, for any function h(x, y), we define Z(A; h(x, y)) as the function

obtained from Z(A) on replacing each variable ai by h(x ~, yi).

P6LYA'S THEOREM. The configuration counting series is obtained by substituting the figure counting series into the cycle index of the configuration group,

C(x, y) = Z(A; c(x, y)). (7)

We return to the necklace example, with D the set of places on the J necklace framework and R = {black, white}. Let black be assigned the

UNSOLVED GRAPHICAL ENUMERATION PROBLEMS, IV 5

weight (1,0) and white the weight (0,1), so that c(x, y ) = x + y. The group D 5 has cycle index

Z(D5 ) = 1 (aln + 4a 5 q_ 5ala2~),

so by P61ya's Theorem,

C(x, y) = Z (Ds , x + y)

1 = 10 ((x + y)5 + 4(x 5 -k yS) + 5(x + y)(x 2 -}- y2)=)

= x 5 + x4y + 2xay2 + 2x~ya + xy4 + yS.

These coefficients are verified in Figure 1. We thus know not only that there are 8 necklaces, but also the number of necklaces with each color combination.

Graphs withp points and q lines can be enumerated in the same manner. Let D = V (m be the set of (~) pairs of points, and R the set {yes, no}, representing the presence or absence of a line joining a given pair of points, Now consider the one-variable form of P61ya's Theorem, C(x) = Z(A; e(x)), and assign "yes" the weight 1 and "no" the weight 0, so that c(x) = 1 + x. The group acting on the domain V (~) is the symmetric pair group S~ ~), which is induced by the symmetric group S~ acting on the set V of p points. P61ya first showed (see [3, 8]) that the cycle index of this group is given by

1 p ! ['/~] (~j) I'aku2~ ] Z(S(~2)) = p l . 1-IP=I Jlc ! kJk k=lH [ --k--l'lJ2k

(8) t -1,,1 E ,,1 )

H -,.+,-'J"+' II 17 . . . . , . , , k=o k=l l<~r~s~'p-1

where d(r, s) and re(r, s) are the greatest common divisor and least common multiple, respectively.

Let g,~ be the number of graphs with p points and q lines, and let

g,(x) = Z g'q xq q

be the counting polynomial for graphs with p points. Then by P61ya's Theorem,

g~(x) = Z g,q xq = Z(S(~ ~), 1 + x). (9) q

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3. OPERATIONS ON PERMUTATION GROUPS AND THE INDUCED ENUMERATION THEOREMS

Further generalizations can be obtained by considering binary operations on permutation groups. In fact, it seems to be a meta-theorem that for every binary operation on permutation groups there is a corresponding enumeration theorem. We now examine three such operations.

Let A be a permutation group of order m = [ A ] and degree d acting on the set X = { x I , x 2 . . . . . x d } , and let B be another permutation group of order n = IB[ and degree e acting on the set Y = { y l , y 2 ..... Ye}- Their s u m A + B is a permutation group which acts on the disjoint union X w Y and whose elements are all the ordered pairs, written as the juxta- position ~/3, of permutations o~ in A and/3 in B. Any element z of X tj Y is permuted by o~fl according to the rule:

I O~Z~ Z ~ X, o~5(z) = ~/3z, z ~ r .

The p r o d u c t A x B of A and B is a permutation group which acts on the set X x Y and whose permutations are all the ordered pairs, written (o~, fi), of permutations a in A and fl in B. The element (x, y) of X x Y is permuted by (~,/3) as expected:

(~, f l ) (x , y ) = (o~x, f ly ) .

The p o w e r g r o u p "B to the A" acts on y x , the set of all functions from X into I1, and is denoted by B A. We will always assume that the power group acts on more than one function. For each pair of permutations

in A and fl in B there is a unique permutation, written (~; fl) in B A. We specify the action of (~; fl) on any function f in y x by the following equation which gives the image of each x ~ X under the function (cq f l ) f :

((cq f l ) f ) ( x ) = f l f (o~x). (14.4)

We summarize this information in the following table:

sum product power

group A B A + B A x B B a objects X Y W u Y X • Y y x

order m n mn mn mn

degree d e d -b e de e a


We remark in passing that the sum, product, and power group are all isomorphic as abstract groups, although of course they are different as permutation groups. For each of these three binary operations, there is a corresponding enumeration theorem. The first of these appeared in P61ya's original paper [11].

SUM GROUP ENUMERATION THEOREM. The cycle index o f the sum of two groups is the product o f the cycle indices o f the groups,

z(A + B) = z(A) z(B). (10)

As an example, consider the number of pairs of necklaces a person could wear, one of length 4 and one of length 5, both with black and white beads. We already know that

Z(D 5 , x § y) = x 5 q- x4y + 2x3y 2 q-2x2y 3 q- xy 4 q- yS.

We find similarly that

Z ( D 4 , x -~ y ) = x 4 -~- x3y -~- 2x2y 2 + xy z + ya.

Then the number of such pairs of necklaces is counted by

Z(D4 q- Ds , x q- y) = Z(D4, x q- y) Z(D5, x q- y)

= x 9 + 2xSy + 5xTy 2 + 7xay 3 -I- 9xSy 4

_5 9x4y5 + 7x3ye q_ 5x2y7 + 2xyS q_ yD.

The theorem corresponding to the product of two groups is somewhat more complex [4].

PRODUCT GROUP ENUMERATION THEOREM. The cycle index o f the product A • B is given by

1 a" ,.(cO.(~) am(r, s) Z(A • B) = I A t I B I ~' ]-I tr.8,,~ ~, (11)

We illustrate this theorem by counting bicolored graphs with 2 points of one color (white) and 3 points of another (black). Let 2" be a set of 2 points and Y a disjoint set of 3 points. The domain then is 2" • Y and the permutation group acting on it is the product $2 • $3 �9 Then by (11) we have,

Z(S3 • 5'2) = 1 (a e + 3a12a2 ~ + 2aa2 + 4 a 3 _q_ 2ae). I Z , ,

8 HARARY

As in counting graphs, the figure series is again 1%- x. After substituting 1%- x i for ai, we obtain the generating function for the bicolored graphs, with 2 points of one color and 3 points of another, enumerated by number of lines:

Z ( S 3 • 2 4 7 1 + x § ~ + 3 x 3%-3x ~%-x 5%-x 6.

The corresponding graphs are shown in Figure 2.

o ~

o o

FIGURE 2

All the examples considered up to now have been cases of counting functions when a permutation group acted on the domain. Using the notation of power groups, we can express the group that acted on the functions in R ~ as E A, where A is the group acting on the domain elements and E is the identity group acting on the range. It is natural to consider what happens when non-trivial groups act on both the domain and the range.

The next result was first found by de Bruijn [1]; the equation (12) below gives the version which appears in [9], where it was discovered that the power group is involved.

POWER GROUP ENUMERATION THEOREM. The number of equivalence classes of functions in R D determined by the power group B ~ is

where

N(Ba ) _ 1 ~ Z(A; m~(~), m2(fl),..., ma(fi)),

m~(~)= Y, sjX~). slk

(12)


As an application of this theorem, we count the number b~ of graphs that are unique up to isomorphism or complementation, so that each graph is equivalent to its complement as well as itself. The group whose orbits are these equivalence classes is the group B s*, where B = S~ 2). Let fix = (1)(2) and fi~ = (1, 2). Then we have

lO, k odd, mk(fil ) = 2 and mk(fi2) = 2, k even.

Substituting these values into (12), we have

N((S(~2))s 0 ---- I[Z(S(~); 2, 2,...) + Z(v(-~); 0, 2,...)1.

From this result one can easily obtain a formula for the number of self-complementary graphs [12]. Indeed, the number c, of self-complementary graphs is

% ---- 2b~ -- g~,

where g~ is the number of graphs with p points, so that

c~ = Z(S~2); 0, 2, 0, 2,...). (13)

The original formulation of this enumeration theorem by de Bruijn [I], although considerably different in appearance, is equivalent to the Power Group Enumeration Theorem. In particular, any problem that can be solved by his method can also be solved by the formula in Harary and Palmer [9] which gives the variable form of (12), just as P61ya's Theorem (7) is the variable form of Burnside's Theorem (1).

4. UNSOLVED GRAPHICAL ENUMERATION PROBLEMS

Our first list of 27 unsolved enumeration problems [5] appeared in 1960. Four years later two of these had been solved, but fortunately two more suggested themselves so that our second list [6] also had 27 problems. Actually by 1967, 6�89 more problems had been solved. In addition, one problem was dropped and one pair of problems was merged. This left the list with only 19 problems but luckily there were 8 more waiting, and so the third list [7, p. 30] also appeared with 27. At this writing two more problems have been solved and others have been merged. The 16 problems that have remained unsolved since 1960 are indicated here with an asterisk.

We include a few definitions needed for list IV which were not in [7]. A 2-lattice is a graph whose points are ordered pairs of integers (i , j) where i = 0, 1 ..... m a n d j = 0, 1 ..... n; two of these points are adjacent whenever their distance in the Cartesian plane is 1. A 3-lattice is defined similarly. An

lO HARARY

even subgraph H of a graph G is one in which every po in t has even degree. Thus every even subgraph of a 2-lattice has a certain area, the number of squares contained in its cycles.

UNSOLVED GRAPHICAL ENUMERATION PROBLEMS, IV

Category Enumerate

Digraph *Strong digraphs *Unilateral digraphs *Digraphs with a source Transitive digraphs Digraphs which are both self-complementary and self-converse

Traversability Hamiltonian graphs Hamiltonian cycles in a given graph Eulerian trails in a given graph

Topologic *Simplicial complexes *k-colorable graphs Planar k-colorable graphs Rooted and edge-rooted plane maps Self-dual planar graphs

Symmetry *Symmetric graphs *Identity graphs *Graphs with given automorphism group

Applications Even subgraphs of a labeled 3-1attice Even subgraphs of a labeled 2-lattice with given area Even subgraphs of a given labeled graph

*Pavings of a 2-lattice *Animals

Miscellaneous *Line graphs *Latin squares *Graphs with given radius or diameter *Graphs with given girth or circumference *Graphs with given connectivity *Graphs with given genus, thickness, chromatic number, etc.

* These problems were in the original list of 27 !

We hope that it will soon be necessary to revise this list again, as solu- t ions are obtained that trivialize quest ions previously appearing extra- ordinari ly difficult, i f no t impossible.

UNSOLVED GRAPHICAL ENUMERATION PROBLEMS, IV l i

REFERENCES

1. N. G. DE BRUUN, Generalization of P61ya's Fundamental Theorem in Enumeration Combinatorial Analysis, Nederl. Akad. Wetensch, Proc. Ser. A 62 (1959), 59-69.

2. W. BURNSIDE, Theory of Groups of Finite Order (2nd ed.), Cambridge University Press, Cambridge, 1911.

3. F. HARARY, The Number of Linear, Directed, Rooted, and Connected Graphs, Trans. Amer. Math. Soc. 78 (1955), 445-463.

4. F. HARARY, On the Number of Bicolored Graphs, Pacific J. Math. 8 (1958), 743-755.

5. F. HARARY, Unsolved Problems in the Enumeration of Graphs, Pubk Math. Inst. Hungar. Acad. Sci. 5 (1960), 63-95.

6. F. HARARY, Combinatorial Problems in Graphical Enumeration, Applied Combi- natorial Mathematics (E. F. Beckenbach, ed.), Wiley, New York, 1964, pp. 185-217.

7. F. HARARY, Graphical Enumeration Problems, Chapter I in Graph Theory and Theoretical Physics (F. Harary, ed.), Academic Press, London/New York, 1967, pp. 1-41.

8. F. HARARY, Enumeration of Graphs and Digraphs, Chapter 6 in A Seminar in Graph Theory (F. Harary, ed.), Holt, Rinehart, & Winston, New York, 1967.

9. F. HARARY AND E. M. PALMER, The Power Group Enumeration Theorem, J. Combinatorial Theory 1 (1966), 157-173.

10o E. M. PALMER AND R. W. ROBINSON, The Matrix Group of Two Permutation Groups, Bull. Amer. Math. Soc. 73 (1967), 204-207.

11. G. POLYA, Kombinatorische Anzahlbestimmungen fiir Gruppen, Graphen und chemische Verbindugen, Acta Math. 68 (1938), 145-254.

12. R. C. READ, On the Number of Self-complementary Graphs and Digraphs, J. London Math. Soc. 38 (1963), 99-104.

13. G.-C. ROTA, On the Foundations of Combinatorial Theory, I: Theory of M6bius Functions, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 2 (1964), 340-368.

enumeration under group action: unsolved graphical enumeration problems, iv

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